Question

Suppose a parallelogram is drawn on a coordinate plane. Using vertices with the coordinates (0,0) , (2,3) , (5,2) , and (3,−1) , compute the perimeter. Round your answer to the nearest tenth.

Answer 1

To calculate the perimeter of the parallelogram defined by the vertices \((0,0)\), \((2,3)\), \((5,2)\), and \((3,-1)\), we need to compute the lengths of the sides.

### Step 1: Calculate the lengths of the sides.

1. **Length between (0,0) and (2,3)**:
\[
d_1 = \sqrt{(2-0)^2 + (3-0)^2} = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13}
\]

2. **Length between (2,3) and (5,2)**:
\[
d_2 = \sqrt{(5-2)^2 + (2-3)^2} = \sqrt{3^2 + (-1)^2} = \sqrt{9 + 1} = \sqrt{10}
\]

3. **Length between (5,2) and (3,-1)**:
\[
d_3 = \sqrt{(3-5)^2 + (-1-2)^2} = \sqrt{(-2)^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13}
\]

4. **Length between (3,-1) and (0,0)**:
\[
d_4 = \sqrt{(0-3)^2 + (0-(-1))^2} = \sqrt{(-3)^2 + 1^2} = \sqrt{9 + 1} = \sqrt{10}
\]

### Step 2: Compute the perimeter.

The perimeter \(P\) of the parallelogram is given by the sum of the lengths of all sides:
\[
P = d_1 + d_2 + d_3 + d_4 = \sqrt{13} + \sqrt{10} + \sqrt{13} + \sqrt{10} = 2\sqrt{13} + 2\sqrt{10}
\]

To find the numerical values, we calculate:
- \(\sqrt{13} \approx 3.6056\)
- \(\sqrt{10} \approx 3.1623\)

Now substituting these values:
\[
P \approx 2(3.6056) + 2(3.1623) = 7.2112 + 6.3246 \approx 13.5358
\]

### Step 3: Round the perimeter to the nearest tenth.

Rounding \(13.5358\) to the nearest tenth gives:
\[
\boxed{13.5}
\]